Faster augmented Lagrangian method with inertial steps for solving convex optimization problems with linear constraints

A faster augmented Lagrangian method (Faster ALM) with constant inertial parameters for solving convex optimization problems with linear equality constraint is presented in this paper. The proposed faster ALM exhibits linear convergence rates in non-ergodic (the last iteration) sense of the Lagrangian primal-dual gap, the objective function value and the feasibility measure. In addition, we prove that the sequence generated by the faster ALM converges to a saddle point of the linear equality constrained optimization problem. This is the first result that provides both linear non-ergodic convergence rate and the convergence of the iterative sequence when solving linearly constrained convex optimization problems by inertial ALM-type algorithms without additional assumptions such as strong convexity or Lipschitz continuous of gradient. As an extension, we present contracting ALM for solving convex optimization problems with linear equality constraint with O (1/& sum;(k)(i=0 )a(i))(a(i )> 0 is an arbitrarily constant) non-ergodic convergence rates of the Lagrangian primal-dual gap, the objective function value and the feasibility measure.